Square free words as products of commutators

TL;DR

The paper demonstrates that for high genus, there exist square-free words in free groups that can be expressed by many distinct Wicks forms, revealing new structural insights into commutator subgroup elements.

Contribution

It introduces a method to construct square-free words with multiple Wicks form representations in free groups for large genus g.

Findings

Existence of sequences of words w(g) with f(g)>g! Wicks form representations

Construction of square-free words with multiple Wicks form decompositions

Results applicable for sufficiently high genus g

Abstract

Elements of the commutator subgroup of a free group can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus g, there is a sequence of words w(g) which can be presented by f(g) distinct Wicks forms, where f(g)>g!. Moreover we may choose these words w(g) to be square free.